Abstract
We give a combinatorial upper bound for the gonality of a curve that is defined by a bivariate Laurent polynomial with given Newton polygon. We conjecture that this bound is generically attained, and provide proofs in a considerable number of special cases. One proof technique uses recent work of M. Baker on linear systems on graphs, by means of which we reduce our conjecture to a purely combinatorial statement.
Article PDF
Similar content being viewed by others
Avoid common mistakes on your manuscript.
References
Baker, H.: Examples of applications of Newton’s polygon to the theory of singular points of algebraic functions. Trans. Camb. Philos. Soc. 15, 403–450 (1893)
Baker, M.: Specialization of linear systems from curves to graphs. Algebra Number Theory 2(6), 613–653 (2008)
Baker, M., Norine, S.: Riemann–Roch and Abel–Jacobi theory on a finite graph. Adv. Math. 215, 766–788 (2007)
Batyrev, V., Nill, B.: Multiples of lattice polytopes without interior lattice points. Mosc. Math. J. 7(2), 195–207 (2007)
Bosma, W., Cannon, J., Playoust, C.: The Magma algebra system I: The user language. J. Symb. Comput. 24, 235–265 (1997)
Beelen, P.: A generalization of Baker’s theorem. Finite Fields Appl. 15(5), 558–568 (2009)
Bruns, W., Gubeladze, J.: Polytopes, Rings, and K-theory. Springer Monographs in Mathematics Springer, Berlin (2009)
Castryck, W., Voight, J.: On nondegeneracy of curves. Algebra Number Theory 3(3), 255–281 (2009)
Cox, D.A.: Toric varieties and toric resolutions. In: Conference Proceedings of ‘Resolution of Singularities’, Obergurgl, 1997. Progress in Mathematics, vol. 181, pp. 259–284 (2000)
Eisenbud, D.: The Geometry of Syzygies: A Second Course in Commutative Algebra and Algebraic Geometry. Graduate Texts in Mathematics, vol. 229. Springer, New York (2005). xvi+243 pp.
Eisenbud, D., Lange, H., Martens, G., Schreyer, F.-O.: The Clifford dimension of a projective curve. Compos. Math. 72, 173–204 (1989)
Gathmann, A., Kerber, M.: A Riemann–Roch theorem in tropical geometry. Math. Z. 259(1), 217–230 (2008)
Gel’fand, I., Kapranov, M., Zelevinsky, A.: Discriminants, Resultants, and Multidimensional Determinants. Mathematics: Theory and Applications. Birkhäuser, Boston (1994)
Green, M.: Koszul cohomology and the geometry of projective varieties. J. Differ. Geom. 19(1), 125–171, 279–289 (1984)
Haase, C., Schicho, J.: Lattice polygons and the number 2i+7. Am. Math. Mon. 116(2), 151–165 (2009)
Hladký, J., Král’, D., Norine, S.: Rank of divisors on tropical curves. Preprint (2007)
Lange, H., Martens, G.: On the gonality sequence of an algebraic curve. Manuscr. Math. doi:10.1007/s00229-011-0475-4 (2011)
Kawaguchi, R.: The gonality conjecture for curves on certain toric varieties. Osaka J. Math. 45, 113–126 (2008)
Kawaguchi, R.: The gonality conjecture for curves on toric surfaces with two ℙ1-fibrations. Saitama Math. J. 27, 35–80 (2010)
Khovanskiĭ, A.G.: Newton polyhedra, and toroidal varieties. Funct. Anal. Appl. 11(4), 289–296 (1978)
Kleiman, S.L., Laksov, D.: Another proof of the existence of special divisors. Acta Math. 132, 163–176 (1974)
Koelman, R.J.: The number of moduli of families of curves on toric surfaces. Ph.D. Thesis, Katholieke Universiteit Nijmegen (1991)
Koelman, R.J.: A criterion for the ideal of a projectively embedded toric surface to be generated by quadrics. Beiträge Algebra Geom. 34(1), 57–62 (1993)
Kouchnirenko, A.G.: Polyèdres de Newton et nombres de Milnor. Invent. Math. 32(1), 1–31 (1976)
Lubbes, N., Schicho, J.: Lattice polygons and families of curves on rational surfaces. J. Algebr. Comb. 34(2), 213–236 (2011)
Luo, Y.: Rank-determining sets of metric graphs. J. Comb. Theory A 118(6) 1775–1793 (2011)
Martens, G.: Über den Clifford Index algebraischer Kurven. J. Reine Angew. Math. 320, 68–85 (1980)
Martens, G.: The gonality of curves on a Hirzebruch surface. Arch. Math. 67(4), 349–352 (1996)
Namba, M.: Families of Meromorphic Functions on Compact Riemann Surfaces. Lecture Notes in Mathematics, vol. 767. Springer, Berlin (1979)
Schreyer, F.O.: Green’s conjecture for the general p-gonal curve of large genus. In: Conference Proceedings of ‘Algebraic Curves and Projective Geometry, Trento 1988, Springer Lecture Notes in Mathematics, vol. 1389, pp. 254–260 (1989)
Schreyer, F.O.: Some topics in computational algebraic geometry. In: Conference Proceedings of ‘Advances in Algebra and Geometry, Hyderabad 2001, pp. 263–278 (2003)
Shustin, E.: A tropical approach to enumerative geometry. St. Petersburg Math. J. 17(2), 343–375 (2006)
Author information
Authors and Affiliations
Corresponding author
Additional information
An erratum to this article can be found at http://dx.doi.org/10.1007/s10801-012-0345-5.
Rights and permissions
About this article
Cite this article
Castryck, W., Cools, F. Newton polygons and curve gonalities. J Algebr Comb 35, 345–366 (2012). https://doi.org/10.1007/s10801-011-0304-6
Received:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s10801-011-0304-6